💡Alright, so I have this problem about time zones and figuring out the best time to communicate with relatives in either Sydney or Los Angeles when it's 18:00 on October 8th in Beijing. Okay, let's break this down step by step.First, I need to understand the time differences provided. The table says Sydney is +2 hours and Los Angeles is -15 hours compared to Beijing time. The positive number for Sydney means it's 2 hours ahead of Beijing, and the negative number for Los Angeles means it's 15 hours behind Beijing.So, when it's 18:00 in Beijing, what time is it in Sydney and Los Angeles?Starting with Sydney: Since Sydney is 2 hours ahead, I just add 2 hours to 18:00. That should be 20:00, which is 8:00 PM in Sydney.Now, for Los Angeles: It's 15 hours behind Beijing. So, I need to subtract 15 hours from 18:00. Let's see, 18:00 minus 15 hours would be 3:00 AM the same day, right? Wait, but if I subtract 15 hours from 18:00, that would actually take me back to the previous day. So, 18:00 minus 12 hours is 6:00 AM, and then minus 3 more hours would be 3:00 AM on October 8th. Wait, no, that doesn't make sense because subtracting 15 hours from 18:00 should be 3:00 AM on October 8th.Wait, but October 8th minus 15 hours would actually be October 7th, wouldn't it? Because 18:00 minus 24 hours would be October 7th at 18:00, so minus 15 hours would be October 7th at 3:00 PM. Hmm, I'm getting confused here. Maybe I should think of it differently.Let me try adding 9 hours to 18:00 to get to the next day. So, 18:00 plus 9 hours is 3:00 AM on October 9th. But that doesn't help me with Los Angeles. Maybe I should use a different approach.I know that Los Angeles is in the Pacific Time Zone, which is UTC-8, and Beijing is UTC+8. So, the difference between them is 16 hours. Wait, but the table says -15 hours. Maybe there's a daylight saving time consideration? I'm not sure.But according to the table, Los Angeles is -15 hours from Beijing. So, if it's 18:00 in Beijing, subtracting 15 hours would be 3:00 AM on October 8th in Los Angeles.Okay, so Sydney is at 20:00 (8:00 PM) and Los Angeles is at 3:00 AM on October 8th.Now, considering communication times, 8:00 PM is a reasonable time to talk in Sydney, but 3:00 AM is very late in Los Angeles. So, Sydney would be a better time to communicate.Wait, but I'm still a bit unsure about the time calculation for Los Angeles. Maybe I should double-check.If Beijing is UTC+8, then Los Angeles, being UTC-8, would be 16 hours behind. So, 18:00 in Beijing minus 16 hours would be 2:00 AM on October 8th in Los Angeles. But the table says -15 hours, so maybe it's considering daylight saving time, making it 15 hours behind instead of 16.In that case, 18:00 minus 15 hours would be 3:00 AM on October 8th in Los Angeles.Okay, that makes sense now. So, Sydney is at 20:00 and Los Angeles is at 3:00 AM.Therefore, Sydney is a better time to communicate.

💡Okay, so I have this problem here about coordinate systems and lines and circles. Let me try to figure it out step by step. First, the problem says that we're taking the origin O of the Cartesian coordinate system as the pole and the positive half-axis of the x-axis as the polar axis. Both systems have equal units of measurement. That part seems straightforward; it's just setting up the coordinate system.It mentions that line l passes through point P(1,1) with an inclination angle α = π/6. So, inclination angle is the angle that the line makes with the positive x-axis, right? So, π/6 is 30 degrees. That should help in determining the slope of the line.Part (1) asks for the parametric equation of line l. Hmm, parametric equations. I remember that parametric equations for a line can be written in terms of a parameter, usually t. The general form is:x = x₀ + at y = y₀ + btwhere (x₀, y₀) is a point on the line, and (a, b) is the direction vector of the line.Since the line passes through P(1,1), that gives us x₀ = 1 and y₀ = 1. Now, we need the direction vector. Since the inclination angle is π/6, the slope of the line is tan(π/6). Let me calculate that.tan(π/6) is 1/√3, which is approximately 0.577. So, the slope is 1/√3. The direction vector can be found from the slope. If the slope is rise over run, which is 1/√3, then the direction vector can be (√3, 1). Because if you go √3 units in the x-direction, you go 1 unit in the y-direction, giving a slope of 1/√3.Wait, let me check that. If the direction vector is (√3, 1), then the slope would be 1/√3, which matches. So, yes, that's correct.Therefore, the parametric equations should be:x = 1 + √3 * t y = 1 + 1 * tBut wait, in the original problem, the parametric equations are written with coefficients as sqrt(3)/2 and 1/2. Hmm, why is that?Oh, right! Because sometimes the direction vector is given as a unit vector. Since the inclination angle is π/6, the direction vector can also be represented using cosine and sine of that angle. So, the direction vector would be (cos(π/6), sin(π/6)).Calculating that:cos(π/6) = √3/2 sin(π/6) = 1/2So, the direction vector is (√3/2, 1/2). Therefore, the parametric equations become:x = 1 + (√3/2) * t y = 1 + (1/2) * tThat makes sense because the direction vector is a unit vector, which might be more convenient in some cases, especially when dealing with distances or when the parameter t represents actual distance along the line.So, for part (1), the parametric equations are:x = 1 + (√3/2) t y = 1 + (1/2) tAlright, that seems solid. I think I got part (1) down.Moving on to part (2). It says that line l intersects the circle ρ = 2 at two points A and B. We need to find the product of the distances from point P to points A and B.First, let me recall that ρ = 2 in polar coordinates represents a circle with radius 2 centered at the origin. So, in Cartesian coordinates, that would be x² + y² = 4.So, we have the line l with parametric equations:x = 1 + (√3/2) t y = 1 + (1/2) tAnd the circle equation is x² + y² = 4.To find the points of intersection A and B, we can substitute the parametric equations into the circle equation and solve for t.Let me write that out:[1 + (√3/2) t]^2 + [1 + (1/2) t]^2 = 4Expanding both squares:First term: [1 + (√3/2) t]^2 = 1 + 2*(√3/2)*t + ( (√3)/2 )² t² = 1 + √3 t + (3/4) t²Second term: [1 + (1/2) t]^2 = 1 + 2*(1/2)*t + (1/2)² t² = 1 + t + (1/4) t²Adding both terms together:1 + √3 t + (3/4) t² + 1 + t + (1/4) t² = 4Combine like terms:1 + 1 = 2 √3 t + t = t(√3 + 1) (3/4)t² + (1/4)t² = t²So, putting it all together:2 + t(√3 + 1) + t² = 4Subtract 4 from both sides:t² + (√3 + 1) t + 2 - 4 = 0 t² + (√3 + 1) t - 2 = 0So, we have a quadratic equation in terms of t:t² + (√3 + 1) t - 2 = 0Let me denote this as:t² + bt + c = 0, where b = √3 + 1 and c = -2We need to find the roots t₁ and t₂ of this quadratic equation. The product of the roots of a quadratic equation at² + bt + c = 0 is c/a. Since a = 1 here, the product t₁ * t₂ = c = -2.But wait, the problem asks for the product of the distances from point P to points A and B. So, |PA| * |PB|.In parametric equations, the parameter t often represents the directed distance from the point P. So, if t₁ and t₂ are the parameters corresponding to points A and B, then |PA| = |t₁| and |PB| = |t₂|.But actually, in parametric equations, t can represent the actual distance if the direction vector is a unit vector. In our case, the direction vector is (√3/2, 1/2), which is indeed a unit vector because (√3/2)² + (1/2)² = 3/4 + 1/4 = 1.Therefore, t represents the actual distance from point P along the line l. So, t₁ and t₂ are the distances from P to A and P to B, but considering direction. However, since we are interested in the product of the distances, which are magnitudes, we can take the absolute value.But wait, in the quadratic equation, t₁ * t₂ = -2. So, |t₁| * |t₂| = |t₁ * t₂| = |-2| = 2.Therefore, the product of the distances from P to A and P to B is 2.Wait, is that correct? Let me think again. So, in parametric equations, t can be positive or negative, representing direction. So, if t₁ and t₂ are the roots, one could be positive and the other negative, meaning one point is on one side of P, and the other is on the opposite side.But when we take the product of the distances, it's the product of the absolute values, so regardless of the sign, it's the magnitude. Therefore, |t₁| * |t₂| = |t₁ * t₂| = | -2 | = 2.Yes, that seems right. So, the product is 2.Alternatively, I remember there's something called the power of a point with respect to a circle. The power of point P with respect to the circle is equal to the product of the lengths of the segments from P to the points of intersection with the circle. The formula is:Power = |PA| * |PB| = |OP|² - r²Wait, no, that's not quite right. The power of a point P with respect to a circle with center O and radius r is defined as:Power = |OP|² - r²But in this case, the circle is centered at O with radius 2, and point P is (1,1). So, |OP| is the distance from O to P, which is sqrt(1² + 1²) = sqrt(2). Therefore, |OP|² = 2.So, the power of point P is |OP|² - r² = 2 - 4 = -2.But wait, power is usually defined as PA * PB, but in this case, since P is inside the circle, the power is negative. However, when we take the product of distances, it's the absolute value. So, |PA| * |PB| = |Power| = | -2 | = 2.Yes, that confirms our earlier result. So, the product is indeed 2.Therefore, the answer to part (2) is 2.I think that's solid. I used two methods: substituting the parametric equations into the circle equation and solving for t, then using Vieta's formula to find the product of the roots, and also using the power of a point theorem. Both methods led me to the same answer, so I feel confident.**Final Answer**The product of the distances from point (P) to points (A) and (B) is boxed{2}.

💡Alright, so I've got this problem here about students borrowing books from the library. Let me try to wrap my head around it step by step. First, there are 50 students in total. Out of these, 4 didn't borrow any books. That's straightforward. Then, 15 students each borrowed 1 book, 9 students borrowed 2 books each, and 7 students borrowed 3 books each. The rest of the students borrowed at least 4 books. The average number of books per student is 3. So, the total number of books borrowed by all 50 students should be 50 times 3, which is 150 books. That makes sense because average is total divided by the number of items, so total is average times the number of items.Now, let's break down the total books borrowed by the students who borrowed 0, 1, 2, or 3 books. - 4 students borrowed 0 books: 4 * 0 = 0 books- 15 students borrowed 1 book each: 15 * 1 = 15 books- 9 students borrowed 2 books each: 9 * 2 = 18 books- 7 students borrowed 3 books each: 7 * 3 = 21 booksAdding these up: 0 + 15 + 18 + 21 = 54 books.So, the remaining books borrowed by the students who borrowed at least 4 books would be 150 total books minus 54 books, which is 96 books. Now, how many students borrowed at least 4 books? The total number of students is 50, and we've accounted for 4 + 15 + 9 + 7 = 35 students. So, 50 - 35 = 15 students borrowed at least 4 books.The problem also states that the total number of books borrowed by those who borrowed at least 4 books must be even. Since 96 is already an even number, that condition is satisfied without any adjustments.Now, the goal is to find the maximum number of books a single student could have borrowed. To maximize one student's books, we need to minimize the number of books the other students in that group borrowed. The minimum number of books a student in this group could have borrowed is 4, as they borrowed at least 4 books.So, if 14 of these 15 students borrowed the minimum of 4 books each, that would be 14 * 4 = 56 books. Subtracting this from the total books borrowed by the group: 96 - 56 = 40 books. Therefore, the maximum number of books a single student could have borrowed is 40. But wait, the problem also mentions that no student borrowed more than 10 books. Hmm, 40 is way more than 10. Did I make a mistake somewhere? Let me double-check. The total books borrowed by the group is 96. If I minimize the books for 14 students, each borrowing 4 books, that's 56 books. Then, the 15th student would have borrowed 96 - 56 = 40 books. But 40 exceeds the maximum limit of 10 books per student. Oh, I see. So, I need to adjust my approach. If no student can borrow more than 10 books, then the maximum any single student can borrow is 10. But wait, the total books borrowed by the group is 96. If I set one student to borrow 10 books, then the remaining 14 students would have borrowed 96 - 10 = 86 books. But 86 books divided by 14 students would be approximately 6.14 books per student, which isn't possible since the number of books must be whole numbers. Alternatively, if I set one student to borrow 10 books, then the remaining 14 students would need to borrow 86 books. To minimize the number of books, I should have as many students as possible borrowing the minimum, which is 4 books. So, 14 students borrowing 4 books each would be 56 books. But 56 is less than 86, so I need to distribute the remaining books among these 14 students. 86 - 56 = 30 books left to distribute. If I add 2 books to 15 students, but wait, there are only 14 students. So, I can add 2 books to 14 students, which would give 14 * 6 = 84 books. But 84 is still less than 86. Wait, maybe I need to adjust differently. If I have 14 students borrowing 4 books each, that's 56. Then, I have 86 - 56 = 30 books left. I can distribute these 30 books among the 14 students. Since each student can borrow up to 10 books, I can add 2 books to 15 students, but there are only 14 students. So, I can add 2 books to 14 students, making their total 6 books each, which is 14 * 6 = 84 books. But 84 is still 2 books short of 86. So, I need to add 2 more books. I can give one student an extra book, making that student borrow 7 books instead of 6. So, 13 students borrow 6 books each, and 1 student borrows 7 books. But wait, 13 * 6 = 78, plus 7 = 85, which is still 1 book short. Alternatively, 14 students borrowing 6 books each is 84, and then I have 2 books left. I can give one student an extra 2 books, making that student borrow 8 books. So, 13 students borrow 6 books each, and 1 student borrows 8 books. But this seems complicated. Maybe there's a better way. Alternatively, instead of setting one student to borrow 10 books, I can set multiple students to borrow more than 4 books but not exceeding 10. But the goal is to maximize the number of books one student borrowed, so I should try to have as many other students as possible borrow the minimum, which is 4 books. So, 14 students borrowing 4 books each is 56 books. Then, the 15th student would borrow 96 - 56 = 40 books, but that's over the limit of 10 books. Therefore, I need to adjust the number of books the other students borrow to ensure that the maximum any student can borrow is 10. Let me try setting one student to borrow 10 books. Then, the remaining 14 students need to borrow 96 - 10 = 86 books. To minimize the number of books the other students borrow, I can have as many as possible borrow 4 books. So, 14 students borrowing 4 books each is 56 books. But 56 is less than 86, so I need to distribute the remaining 86 - 56 = 30 books among these 14 students. Each additional book per student increases the total by 1. So, to distribute 30 books among 14 students, I can add 2 books to 15 students, but there are only 14 students. So, I can add 2 books to 14 students, making their total 6 books each, which is 14 * 6 = 84 books. But 84 is still 2 books short of 86. So, I need to add 2 more books. I can give one student an extra 2 books, making that student borrow 8 books. So, 13 students borrow 6 books each, and 1 student borrows 8 books. But wait, 13 * 6 = 78, plus 8 = 86. So, the distribution would be: - 1 student borrows 10 books- 1 student borrows 8 books- 13 students borrow 6 books eachBut this seems a bit uneven. Is there a way to make it more balanced? Alternatively, I can have some students borrow 7 books each. If I have 14 students borrowing 6 books each, that's 84 books. Then, I need 2 more books to reach 86. So, I can have one student borrow 8 books instead of 6. So, the distribution would be: - 1 student borrows 10 books- 1 student borrows 8 books- 13 students borrow 6 books eachThis adds up to 10 + 8 + (13 * 6) = 10 + 8 + 78 = 96 books. Wait, that's correct. But the maximum number of books a single student could have borrowed is 10, but in this case, one student borrowed 10, another borrowed 8, and the rest borrowed 6. But the question asks for the maximum number of books a single student could have borrowed, given that the total number of books borrowed by those who borrowed at least 4 books must be even. In this scenario, the maximum is 10 books. But earlier, without considering the 10-book limit, the maximum could have been 40 books. But since there's a limit of 10 books per student, the maximum is 10. Wait, but the problem says "no student borrowed more than 10 books," which means that the maximum any student can borrow is 10. Therefore, the maximum number of books a single student could have borrowed is 10. But let me double-check. If I set one student to borrow 10 books, then the remaining 14 students need to borrow 86 books. If I have 14 students borrowing 6 books each, that's 84 books, leaving 2 books. I can't have a student borrow 2 books because they are supposed to borrow at least 4 books. Wait, that's a problem. So, if I have 14 students borrowing 6 books each, that's 84 books, and I still have 2 books left to distribute. But I can't give 2 books to any student because they must borrow at least 4 books. Therefore, I need to adjust the distribution. Instead of having 14 students borrow 6 books each, I can have 13 students borrow 6 books each, which is 78 books, and then have 1 student borrow 8 books (78 + 8 = 86). So, the distribution would be: - 1 student borrows 10 books- 1 student borrows 8 books- 13 students borrow 6 books eachThis adds up to 10 + 8 + (13 * 6) = 10 + 8 + 78 = 96 books. This way, all students who borrowed at least 4 books have borrowed between 6 and 10 books, and the total is 96, which is even. Therefore, the maximum number of books a single student could have borrowed is 10. But wait, is there a way to have one student borrow more than 10 books? The problem says no student borrowed more than 10 books, so 10 is the maximum. Therefore, the answer should be 10 books. But earlier, without considering the 10-book limit, the maximum could have been 40 books. But since there's a limit, it's 10. So, the maximum number of books a single student could have borrowed is 10.

💡Okay, so I have this problem where I need to find ( f(f(f(f(2+i)))) ) for a function ( f(z) ) that's defined differently depending on whether ( z ) is a real number or not. The function is given as:[f(z) = left{begin{array}{cl}z^{3} & text{if } z text{ is real}, z^2 - 1 & text{if } z text{ is not real}.end{array}right.]Alright, so the starting point is ( z = 2 + i ). Since ( 2 + i ) is a complex number and not purely real, I'll use the second part of the function definition, which is ( z^2 - 1 ). Let me compute that step by step.First, I need to calculate ( f(2 + i) ). Since ( 2 + i ) is not real, I use ( f(z) = z^2 - 1 ).Calculating ( (2 + i)^2 ):- ( (2 + i)^2 = 2^2 + 2 cdot 2 cdot i + i^2 = 4 + 4i + (-1) ) because ( i^2 = -1 ).- So, ( 4 + 4i - 1 = 3 + 4i ).Then, subtract 1: ( 3 + 4i - 1 = 2 + 4i ). Wait, hold on, that doesn't seem right. Let me check my calculation again.Wait, no, actually, the function is ( z^2 - 1 ), so it's ( (2 + i)^2 - 1 ). So, I first compute ( (2 + i)^2 ) as above, which is ( 3 + 4i ), and then subtract 1, so ( 3 + 4i - 1 = 2 + 4i ). Hmm, but in the initial problem, the user had ( f(2 + i) = 3 + 4i ). Did I make a mistake?Wait, let me recalculate ( (2 + i)^2 ). It should be ( (2)^2 + 2 cdot 2 cdot i + (i)^2 = 4 + 4i + (-1) = 3 + 4i ). Then subtract 1: ( 3 + 4i - 1 = 2 + 4i ). So, actually, the user's initial calculation was wrong. It should be ( 2 + 4i ), not ( 3 + 4i ). Hmm, maybe I need to double-check that.Wait, no, hold on. If ( f(z) = z^2 - 1 ), then ( f(2 + i) = (2 + i)^2 - 1 ). So, ( (2 + i)^2 = 4 + 4i + i^2 = 4 + 4i - 1 = 3 + 4i ). Then subtract 1: ( 3 + 4i - 1 = 2 + 4i ). So, yes, it should be ( 2 + 4i ), not ( 3 + 4i ). So, the initial user's calculation was incorrect. That might affect the subsequent steps.Wait, but in the initial problem, the user had ( f(2 + i) = 3 + 4i ), then ( f(3 + 4i) = -8 + 24i ), and so on. But if ( f(2 + i) ) is actually ( 2 + 4i ), then the next step would be different.Let me proceed step by step to ensure I don't make the same mistake.1. Compute ( f(2 + i) ): - Since ( 2 + i ) is not real, ( f(z) = z^2 - 1 ). - ( (2 + i)^2 = 4 + 4i + i^2 = 4 + 4i - 1 = 3 + 4i ). - Subtract 1: ( 3 + 4i - 1 = 2 + 4i ). - So, ( f(2 + i) = 2 + 4i ).2. Compute ( f(f(2 + i)) = f(2 + 4i) ): - ( 2 + 4i ) is not real, so again ( f(z) = z^2 - 1 ). - Compute ( (2 + 4i)^2 ): - ( (2)^2 = 4 ) - ( 2 cdot 2 cdot 4i = 16i ) - ( (4i)^2 = 16i^2 = -16 ) - So, ( 4 + 16i - 16 = -12 + 16i ) - Subtract 1: ( -12 + 16i - 1 = -13 + 16i ) - So, ( f(2 + 4i) = -13 + 16i )3. Compute ( f(f(f(2 + i))) = f(-13 + 16i) ): - ( -13 + 16i ) is not real, so ( f(z) = z^2 - 1 ). - Compute ( (-13 + 16i)^2 ): - ( (-13)^2 = 169 ) - ( 2 cdot (-13) cdot 16i = -416i ) - ( (16i)^2 = 256i^2 = -256 ) - So, ( 169 - 416i - 256 = (169 - 256) - 416i = -87 - 416i ) - Subtract 1: ( -87 - 416i - 1 = -88 - 416i ) - So, ( f(-13 + 16i) = -88 - 416i )4. Compute ( f(f(f(f(2 + i)))) = f(-88 - 416i) ): - ( -88 - 416i ) is not real, so ( f(z) = z^2 - 1 ). - Compute ( (-88 - 416i)^2 ): - ( (-88)^2 = 7744 ) - ( 2 cdot (-88) cdot (-416i) = 2 cdot 88 cdot 416i = 73, 88 * 416 is... let me compute 88*400=35,200 and 88*16=1,408, so total 35,200 + 1,408 = 36,608. So, 2*36,608i = 73,216i - ( (-416i)^2 = (416)^2 i^2 = 173,056 * (-1) = -173,056 ) - So, adding them up: 7744 + 73,216i - 173,056 - Compute the real parts: 7744 - 173,056 = -165,312 - So, total is ( -165,312 + 73,216i ) - Subtract 1: ( -165,312 + 73,216i - 1 = -165,313 + 73,216i ) - So, ( f(-88 - 416i) = -165,313 + 73,216i )Wait, that seems quite different from the initial user's answer. Let me check my calculations again, especially the squaring part, because that's where errors often occur.Starting again with ( f(2 + i) ):- ( (2 + i)^2 = 4 + 4i + i^2 = 4 + 4i -1 = 3 + 4i )- Subtract 1: ( 3 + 4i -1 = 2 + 4i ). So, correct.Next, ( f(2 + 4i) ):- ( (2 + 4i)^2 = 4 + 16i + 16i^2 = 4 + 16i -16 = -12 + 16i )- Subtract 1: ( -12 + 16i -1 = -13 + 16i ). Correct.Next, ( f(-13 + 16i) ):- ( (-13 + 16i)^2 = (-13)^2 + 2*(-13)*(16i) + (16i)^2 = 169 - 416i + 256i^2 = 169 - 416i -256 = -87 - 416i )- Subtract 1: ( -87 - 416i -1 = -88 - 416i ). Correct.Finally, ( f(-88 - 416i) ):- ( (-88 - 416i)^2 = (-88)^2 + 2*(-88)*(-416i) + (-416i)^2 = 7744 + 73,216i + 173,056i^2 )- Since ( i^2 = -1 ), this becomes ( 7744 + 73,216i -173,056 )- Compute real parts: ( 7744 - 173,056 = -165,312 )- So, total is ( -165,312 + 73,216i )- Subtract 1: ( -165,312 + 73,216i -1 = -165,313 + 73,216i )So, the final result is ( -165,313 + 73,216i ).Wait, but in the initial problem, the user had a different result. Let me see where the discrepancy is. The user had:- ( f(2 + i) = 3 + 4i )- ( f(3 + 4i) = -8 + 24i )- ( f(-8 + 24i) = -512 + 384i )- ( f(-512 + 384i) = 114,688 + 393,216i )But according to my calculations, the first step should be ( 2 + 4i ), leading to a different result. So, the initial user made a mistake in the first step, which cascaded through the rest. Therefore, my corrected answer is ( -165,313 + 73,216i ).But let me double-check the squaring of ( -88 - 416i ):Compute ( (-88 - 416i)^2 ):- First, square the real part: ( (-88)^2 = 7744 )- Then, the cross term: ( 2 * (-88) * (-416i) = 2 * 88 * 416i = 73,216i )- Then, square the imaginary part: ( (-416i)^2 = (416)^2 * i^2 = 173,056 * (-1) = -173,056 )- Add them all together: ( 7744 + 73,216i -173,056 = (7744 - 173,056) + 73,216i = -165,312 + 73,216i )- Subtract 1: ( -165,312 + 73,216i -1 = -165,313 + 73,216i )Yes, that seems correct. Therefore, the final answer should be ( -165,313 + 73,216i ).But wait, let me check if I made a mistake in the cross term calculation. The cross term is ( 2 * (-88) * (-416i) ). Since both are negative, the product is positive. So, 2 * 88 * 416i.Compute 88 * 416:- 80 * 416 = 33,280- 8 * 416 = 3,328- Total: 33,280 + 3,328 = 36,608- Then, 2 * 36,608 = 73,216i. Correct.So, the cross term is indeed 73,216i. Therefore, the calculation is correct.Thus, the final result is ( -165,313 + 73,216i ).

💡Alright, so I've got this problem here: positive real numbers ( a, b, c ) satisfy the equation ( (a + c)(b^2 + ac) = 4a ). I need to find the maximum value of ( b + c ). Hmm, okay. Let me try to unpack this step by step.First, let me write down the given equation again to make sure I have it right:[ (a + c)(b^2 + ac) = 4a ]So, we have three variables here: ( a, b, c ), all positive real numbers. The equation involves a product of two terms, ( (a + c) ) and ( (b^2 + ac) ), equaling ( 4a ). My goal is to find the maximum possible value of ( b + c ).Hmm, okay. Since all variables are positive, maybe I can manipulate the equation to express some variables in terms of others or find some relationship between them.Let me start by expanding the left-hand side of the equation:[ (a + c)(b^2 + ac) = a(b^2 + ac) + c(b^2 + ac) ][ = ab^2 + a^2c + cb^2 + ac^2 ]So, the equation becomes:[ ab^2 + a^2c + cb^2 + ac^2 = 4a ]Hmm, that's a bit messy. Maybe I can factor some terms or find a way to simplify this.Looking at the terms, I notice that ( ab^2 ) and ( cb^2 ) both have ( b^2 ) as a factor. Similarly, ( a^2c ) and ( ac^2 ) both have ( ac ) as a factor. Let me factor those out:[ b^2(a + c) + ac(a + c) = 4a ]Oh, interesting! So, both terms have ( (a + c) ) as a common factor. Let me factor that out:[ (a + c)(b^2 + ac) = 4a ]Wait, that's just the original equation. Hmm, so factoring didn't really help me simplify it further. Maybe I need a different approach.Let me think about what I'm trying to maximize: ( b + c ). Maybe I can express ( c ) in terms of ( a ) and ( b ), or find some relationship between ( b ) and ( c ) that can help me maximize their sum.Let me try to solve the equation for ( c ). Starting from:[ (a + c)(b^2 + ac) = 4a ]Let me expand this again:[ ab^2 + a^2c + cb^2 + ac^2 = 4a ]Hmm, this seems complicated. Maybe I can rearrange terms to solve for ( c ). Let's see:Bring all terms to one side:[ ab^2 + a^2c + cb^2 + ac^2 - 4a = 0 ]Hmm, this is a quadratic in terms of ( c ), but it's also mixed with ( a ) and ( b ). Maybe this isn't the best way to go.Alternatively, perhaps I can use some inequality to bound ( b + c ). Since all variables are positive, maybe Cauchy-Schwarz or AM-GM inequality can help.Let me recall the AM-GM inequality: for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. Maybe I can apply that somewhere here.Looking back at the equation:[ (a + c)(b^2 + ac) = 4a ]I wonder if I can find a way to relate ( b + c ) to this equation. Maybe I can express ( b^2 ) in terms of ( b + c ). Wait, ( b^2 ) is just ( b times b ), but I don't see a direct connection to ( c ).Alternatively, maybe I can consider ( b + c ) as a single variable, say ( k ), and try to express the equation in terms of ( k ). Let me try that.Let ( k = b + c ). Then, ( c = k - b ). Let me substitute this into the equation:[ (a + (k - b))(b^2 + a(k - b)) = 4a ]Simplify the terms inside the parentheses:First term: ( a + k - b )Second term: ( b^2 + ak - ab )So, the equation becomes:[ (a + k - b)(b^2 + ak - ab) = 4a ]Hmm, this seems more complicated. Maybe this substitution isn't helpful.Let me think differently. Since I need to maximize ( b + c ), perhaps I can fix ( a ) and try to maximize ( b + c ) given ( a ). Or maybe fix ( b ) and ( c ) and see how ( a ) behaves.Alternatively, maybe I can use Lagrange multipliers, treating this as an optimization problem with constraint. But that might be overkill for this problem, and I'm not sure if it's necessary.Wait, another thought: since all variables are positive, maybe I can set ( a = 1 ) to simplify the equation and see what happens. If I can find a relationship when ( a = 1 ), maybe I can generalize.Let me try that. Let ( a = 1 ). Then the equation becomes:[ (1 + c)(b^2 + c) = 4 ]So,[ (1 + c)(b^2 + c) = 4 ]Let me expand this:[ b^2(1 + c) + c(1 + c) = 4 ][ b^2 + b^2c + c + c^2 = 4 ]Hmm, still complicated. Maybe I can solve for ( b^2 ):[ b^2(1 + c) = 4 - c - c^2 ][ b^2 = frac{4 - c - c^2}{1 + c} ]Since ( b^2 ) must be positive, the numerator and denominator must have the same sign. Since ( a, b, c ) are positive, ( 1 + c > 0 ), so the numerator must also be positive:[ 4 - c - c^2 > 0 ][ -c^2 - c + 4 > 0 ][ c^2 + c - 4 < 0 ]Solving the quadratic inequality ( c^2 + c - 4 < 0 ):The roots are ( c = frac{-1 pm sqrt{1 + 16}}{2} = frac{-1 pm sqrt{17}}{2} ). Since ( c ) is positive, we consider the positive root:[ c < frac{-1 + sqrt{17}}{2} approx frac{-1 + 4.123}{2} approx 1.5615 ]So, ( c ) must be less than approximately 1.5615.Now, ( b^2 = frac{4 - c - c^2}{1 + c} ). Let me write this as:[ b^2 = frac{-(c^2 + c - 4)}{1 + c} ]Hmm, not sure if that helps. Maybe I can express ( b ) in terms of ( c ):[ b = sqrt{frac{4 - c - c^2}{1 + c}} ]So, ( b + c = sqrt{frac{4 - c - c^2}{1 + c}} + c ). Now, I can consider this as a function of ( c ) and try to find its maximum.Let me denote ( f(c) = sqrt{frac{4 - c - c^2}{1 + c}} + c ). To find its maximum, I can take the derivative and set it to zero.But this might get complicated. Let me see if I can find critical points or analyze the behavior.First, let's note the domain of ( c ). From earlier, ( c < frac{-1 + sqrt{17}}{2} approx 1.5615 ), and ( c > 0 ).So, ( c in (0, frac{-1 + sqrt{17}}{2}) ).Let me compute ( f(c) ) at some points to get an idea.At ( c = 0 ):[ f(0) = sqrt{frac{4 - 0 - 0}{1 + 0}} + 0 = sqrt{4} = 2 ]At ( c = 1 ):[ f(1) = sqrt{frac{4 - 1 - 1}{1 + 1}} + 1 = sqrt{frac{2}{2}} + 1 = 1 + 1 = 2 ]At ( c = 1.5 ):First, check if ( c = 1.5 ) is within the domain. ( 1.5 < 1.5615 ), so yes.Compute numerator: ( 4 - 1.5 - (1.5)^2 = 4 - 1.5 - 2.25 = 0.25 )Denominator: ( 1 + 1.5 = 2.5 )So,[ f(1.5) = sqrt{frac{0.25}{2.5}} + 1.5 = sqrt{0.1} + 1.5 approx 0.316 + 1.5 = 1.816 ]Hmm, less than 2.At ( c ) approaching ( frac{-1 + sqrt{17}}{2} approx 1.5615 ):Numerator approaches 0, so ( f(c) ) approaches ( c approx 1.5615 ), which is less than 2.So, from these points, it seems that ( f(c) ) reaches a maximum of 2 at ( c = 0 ) and ( c = 1 ). But wait, at ( c = 0 ), ( b = 2 ), so ( b + c = 2 + 0 = 2 ). At ( c = 1 ), ( b = 1 ), so ( b + c = 1 + 1 = 2 ).Hmm, interesting. So, in both cases, the maximum is 2. Maybe 2 is indeed the maximum value of ( b + c ).But wait, let me check another point. Let's try ( c = 0.5 ):Numerator: ( 4 - 0.5 - 0.25 = 3.25 )Denominator: ( 1 + 0.5 = 1.5 )So,[ f(0.5) = sqrt{frac{3.25}{1.5}} + 0.5 approx sqrt{2.1667} + 0.5 approx 1.472 + 0.5 = 1.972 ]Still less than 2.Another point, ( c = 0.25 ):Numerator: ( 4 - 0.25 - 0.0625 = 3.6875 )Denominator: ( 1.25 )So,[ f(0.25) = sqrt{frac{3.6875}{1.25}} + 0.25 approx sqrt{2.95} + 0.25 approx 1.717 + 0.25 = 1.967 ]Still less than 2.Hmm, so it seems that the maximum value of ( b + c ) is 2, achieved when ( c = 0 ) and ( b = 2 ), or when ( c = 1 ) and ( b = 1 ).But wait, when ( c = 0 ), the original equation becomes:[ (a + 0)(b^2 + a cdot 0) = 4a ][ a cdot b^2 = 4a ][ b^2 = 4 ][ b = 2 ]So, that's valid.When ( c = 1 ), and ( a = 1 ), as I set earlier, we have:[ (1 + 1)(b^2 + 1 cdot 1) = 4 ][ 2(b^2 + 1) = 4 ][ b^2 + 1 = 2 ][ b^2 = 1 ][ b = 1 ]So, that's also valid.But wait, in the case when ( a = 1 ), ( c = 1 ), ( b = 1 ), ( b + c = 2 ). Similarly, when ( a = 1 ), ( c = 0 ), ( b = 2 ), ( b + c = 2 ).But what if ( a ) isn't 1? Does the maximum value of ( b + c ) still hold at 2?Let me check with a different ( a ). Let's say ( a = 2 ). Then the equation becomes:[ (2 + c)(b^2 + 2c) = 8 ]Let me try to find ( b + c ) in this case.Let me assume ( b + c = 2 ). Then ( c = 2 - b ). Substitute into the equation:[ (2 + (2 - b))(b^2 + 2(2 - b)) = 8 ][ (4 - b)(b^2 + 4 - 2b) = 8 ]Let me expand this:First, multiply ( (4 - b) ) with ( b^2 ):[ 4b^2 - b^3 ]Then, multiply ( (4 - b) ) with ( 4 ):[ 16 - 4b ]Then, multiply ( (4 - b) ) with ( -2b ):[ -8b + 2b^2 ]So, combining all terms:[ 4b^2 - b^3 + 16 - 4b - 8b + 2b^2 = 8 ]Combine like terms:- ( b^3 ): ( -b^3 )- ( b^2 ): ( 4b^2 + 2b^2 = 6b^2 )- ( b ): ( -4b - 8b = -12b )- Constants: ( 16 )So, the equation becomes:[ -b^3 + 6b^2 - 12b + 16 = 8 ]Subtract 8 from both sides:[ -b^3 + 6b^2 - 12b + 8 = 0 ]Multiply both sides by -1:[ b^3 - 6b^2 + 12b - 8 = 0 ]Let me try to factor this cubic equation. Maybe ( b = 2 ) is a root:[ 2^3 - 6(2)^2 + 12(2) - 8 = 8 - 24 + 24 - 8 = 0 ]Yes, ( b = 2 ) is a root. So, we can factor out ( (b - 2) ):Using polynomial division or synthetic division:Divide ( b^3 - 6b^2 + 12b - 8 ) by ( b - 2 ):- Coefficients: 1 | -6 | 12 | -8- Bring down 1- Multiply by 2: 1*2 = 2- Add to next coefficient: -6 + 2 = -4- Multiply by 2: -4*2 = -8- Add to next coefficient: 12 + (-8) = 4- Multiply by 2: 4*2 = 8- Add to last coefficient: -8 + 8 = 0So, the cubic factors as:[ (b - 2)(b^2 - 4b + 4) = 0 ]Which further factors as:[ (b - 2)(b - 2)^2 = 0 ][ (b - 2)^3 = 0 ]So, the only real root is ( b = 2 ). Therefore, when ( a = 2 ), ( b = 2 ), and ( c = 0 ), the equation holds, and ( b + c = 2 ).So, even when ( a = 2 ), the maximum ( b + c ) is 2.Hmm, interesting. Let me try another value, say ( a = 0.5 ).Then the equation becomes:[ (0.5 + c)(b^2 + 0.5c) = 2 ]Assume ( b + c = 2 ), so ( c = 2 - b ). Substitute:[ (0.5 + (2 - b))(b^2 + 0.5(2 - b)) = 2 ]Simplify:First term: ( 0.5 + 2 - b = 2.5 - b )Second term: ( b^2 + 1 - 0.5b )So, the equation becomes:[ (2.5 - b)(b^2 + 1 - 0.5b) = 2 ]Let me expand this:Multiply ( 2.5 ) with each term in the second factor:- ( 2.5 times b^2 = 2.5b^2 )- ( 2.5 times 1 = 2.5 )- ( 2.5 times (-0.5b) = -1.25b )Multiply ( -b ) with each term in the second factor:- ( -b times b^2 = -b^3 )- ( -b times 1 = -b )- ( -b times (-0.5b) = 0.5b^2 )Combine all terms:[ 2.5b^2 + 2.5 - 1.25b - b^3 - b + 0.5b^2 = 2 ]Combine like terms:- ( b^3 ): ( -b^3 )- ( b^2 ): ( 2.5b^2 + 0.5b^2 = 3b^2 )- ( b ): ( -1.25b - b = -2.25b )- Constants: ( 2.5 )So, the equation becomes:[ -b^3 + 3b^2 - 2.25b + 2.5 = 2 ]Subtract 2 from both sides:[ -b^3 + 3b^2 - 2.25b + 0.5 = 0 ]Multiply both sides by -1:[ b^3 - 3b^2 + 2.25b - 0.5 = 0 ]Let me try ( b = 1 ):[ 1 - 3 + 2.25 - 0.5 = -0.25 neq 0 ]Hmm, not zero. Try ( b = 0.5 ):[ 0.125 - 0.75 + 1.125 - 0.5 = 0 ]Wait, let me compute:( 0.5^3 = 0.125 )( -3(0.5)^2 = -0.75 )( 2.25(0.5) = 1.125 )( -0.5 )So,[ 0.125 - 0.75 + 1.125 - 0.5 = (0.125 + 1.125) + (-0.75 - 0.5) = 1.25 - 1.25 = 0 ]Yes, ( b = 0.5 ) is a root. So, we can factor out ( (b - 0.5) ):Using synthetic division:Coefficients: 1 | -3 | 2.25 | -0.5Root: 0.5Bring down 1.Multiply by 0.5: 1 * 0.5 = 0.5Add to next coefficient: -3 + 0.5 = -2.5Multiply by 0.5: -2.5 * 0.5 = -1.25Add to next coefficient: 2.25 + (-1.25) = 1Multiply by 0.5: 1 * 0.5 = 0.5Add to last coefficient: -0.5 + 0.5 = 0So, the cubic factors as:[ (b - 0.5)(b^2 - 2.5b + 1) = 0 ]Now, solve ( b^2 - 2.5b + 1 = 0 ):Discriminant: ( (2.5)^2 - 4(1)(1) = 6.25 - 4 = 2.25 )Roots:[ b = frac{2.5 pm sqrt{2.25}}{2} = frac{2.5 pm 1.5}{2} ]So,[ b = frac{2.5 + 1.5}{2} = 2 ][ b = frac{2.5 - 1.5}{2} = 0.5 ]So, the roots are ( b = 0.5 ) and ( b = 2 ). Therefore, when ( a = 0.5 ), ( b = 0.5 ) or ( b = 2 ). If ( b = 0.5 ), then ( c = 2 - 0.5 = 1.5 ). If ( b = 2 ), then ( c = 0 ).Check if these satisfy the original equation:For ( b = 0.5 ), ( c = 1.5 ):[ (0.5 + 1.5)(0.5^2 + 0.5 times 1.5) = (2)(0.25 + 0.75) = 2 times 1 = 2 ]Which equals ( 4a = 4 times 0.5 = 2 ). So, valid.For ( b = 2 ), ( c = 0 ):[ (0.5 + 0)(2^2 + 0.5 times 0) = 0.5 times 4 = 2 ]Which also equals ( 4a = 2 ). So, valid.So, in both cases, ( b + c = 2 ).Hmm, so regardless of the value of ( a ), it seems that the maximum value of ( b + c ) is 2. Whether ( a ) is 0.5, 1, 2, or any positive real number, the maximum ( b + c ) is 2.But wait, is this always the case? Let me try another value, say ( a = 3 ).Then the equation becomes:[ (3 + c)(b^2 + 3c) = 12 ]Assume ( b + c = 2 ), so ( c = 2 - b ). Substitute:[ (3 + (2 - b))(b^2 + 3(2 - b)) = 12 ]Simplify:First term: ( 5 - b )Second term: ( b^2 + 6 - 3b )So, the equation becomes:[ (5 - b)(b^2 + 6 - 3b) = 12 ]Let me expand this:Multiply ( 5 ) with each term in the second factor:- ( 5 times b^2 = 5b^2 )- ( 5 times 6 = 30 )- ( 5 times (-3b) = -15b )Multiply ( -b ) with each term in the second factor:- ( -b times b^2 = -b^3 )- ( -b times 6 = -6b )- ( -b times (-3b) = 3b^2 )Combine all terms:[ 5b^2 + 30 - 15b - b^3 - 6b + 3b^2 = 12 ]Combine like terms:- ( b^3 ): ( -b^3 )- ( b^2 ): ( 5b^2 + 3b^2 = 8b^2 )- ( b ): ( -15b - 6b = -21b )- Constants: ( 30 )So, the equation becomes:[ -b^3 + 8b^2 - 21b + 30 = 12 ]Subtract 12 from both sides:[ -b^3 + 8b^2 - 21b + 18 = 0 ]Multiply both sides by -1:[ b^3 - 8b^2 + 21b - 18 = 0 ]Let me try ( b = 2 ):[ 8 - 32 + 42 - 18 = 0 ]Yes, ( 8 - 32 = -24 ), ( -24 + 42 = 18 ), ( 18 - 18 = 0 ). So, ( b = 2 ) is a root.Factor out ( (b - 2) ):Using synthetic division:Coefficients: 1 | -8 | 21 | -18Root: 2Bring down 1.Multiply by 2: 1 * 2 = 2Add to next coefficient: -8 + 2 = -6Multiply by 2: -6 * 2 = -12Add to next coefficient: 21 + (-12) = 9Multiply by 2: 9 * 2 = 18Add to last coefficient: -18 + 18 = 0So, the cubic factors as:[ (b - 2)(b^2 - 6b + 9) = 0 ]Which further factors as:[ (b - 2)(b - 3)^2 = 0 ]So, roots are ( b = 2 ) and ( b = 3 ).If ( b = 2 ), then ( c = 0 ). Check:[ (3 + 0)(2^2 + 3 times 0) = 3 times 4 = 12 ]Which equals ( 4a = 12 ). Valid.If ( b = 3 ), then ( c = 2 - 3 = -1 ). But ( c ) must be positive, so this is invalid.So, only ( b = 2 ), ( c = 0 ) is valid, giving ( b + c = 2 ).Hmm, so again, the maximum ( b + c ) is 2.Wait a second, so no matter what value ( a ) takes, as long as it's positive, the maximum value of ( b + c ) is 2. That seems consistent across different values of ( a ).But let me think about this more generally. Maybe I can prove that ( b + c leq 2 ) using the given equation.Starting from the given equation:[ (a + c)(b^2 + ac) = 4a ]Let me try to manipulate this inequality to find an upper bound for ( b + c ).First, note that all variables are positive, so ( a + c > 0 ) and ( b^2 + ac > 0 ).Let me consider the expression ( b + c ). Maybe I can relate it to the given equation.Let me try to apply the AM-GM inequality on ( b^2 ) and ( ac ):[ frac{b^2 + ac}{2} geq sqrt{b^2 cdot ac} = b sqrt{ac} ]But I'm not sure if this helps directly.Alternatively, maybe I can use Cauchy-Schwarz inequality on the terms ( (a + c) ) and ( (b^2 + ac) ).Wait, Cauchy-Schwarz in the form ( (x_1y_1 + x_2y_2)^2 leq (x_1^2 + x_2^2)(y_1^2 + y_2^2) ). Not sure if that applies here.Alternatively, maybe I can use the inequality ( (a + c)(b^2 + ac) geq (sqrt{a b^2} + sqrt{c cdot ac})^2 ). Let me check:Using the Cauchy-Schwarz inequality for two terms:[ (a + c)(b^2 + ac) geq (sqrt{a b^2} + sqrt{c cdot ac})^2 ][ = (b sqrt{a} + c sqrt{a})^2 ][ = a(b + c)^2 ]So, we have:[ (a + c)(b^2 + ac) geq a(b + c)^2 ]But from the given equation, ( (a + c)(b^2 + ac) = 4a ). Therefore,[ 4a geq a(b + c)^2 ]Since ( a > 0 ), we can divide both sides by ( a ):[ 4 geq (b + c)^2 ]Taking square roots (since ( b + c > 0 )):[ 2 geq b + c ]So, ( b + c leq 2 ).Wow, that's a neat application of the Cauchy-Schwarz inequality! So, regardless of the values of ( a, b, c ), as long as they satisfy the given equation, ( b + c ) cannot exceed 2.And earlier, we saw that ( b + c = 2 ) is achievable when ( c = 0 ) and ( b = 2 ), or when ( c = 1 ) and ( b = 1 ), depending on the value of ( a ).Therefore, the maximum value of ( b + c ) is indeed 2.**Final Answer**The maximum value of ( b + c ) is boxed{2}.

💡Okay, so I have this problem with three equations involving positive integers a, b, c, d, and e. The equations are:1. ((a + 1)(3bc + 1) = d + 3e + 1)2. ((b + 1)(3ca + 1) = 3d + e + 13)3. ((c + 1)(3ab + 1) = 4(26 - d - e) - 1)And I need to find (d^2 + e^2). Hmm, okay, let's see how to approach this.First, since all variables are positive integers, maybe I can find some relationships or constraints that can help me narrow down the possible values of a, b, c, d, and e.Looking at the equations, they all have a similar structure. Each equation is of the form ((variable + 1)(3 * product of the other two variables + 1)) equals some linear combination of d and e plus a constant. Maybe if I add all three equations together, I can find a relationship that simplifies things.Let me try that. Adding the left-hand sides:((a + 1)(3bc + 1) + (b + 1)(3ca + 1) + (c + 1)(3ab + 1))And the right-hand sides:((d + 3e + 1) + (3d + e + 13) + [4(26 - d - e) - 1])Let me compute the right-hand side first because it seems manageable. Let's expand each term:1. (d + 3e + 1)2. (3d + e + 13)3. (4(26 - d - e) - 1 = 104 - 4d - 4e - 1 = 103 - 4d - 4e)Now, adding them all together:(d + 3e + 1 + 3d + e + 13 + 103 - 4d - 4e)Combine like terms:- For d: (d + 3d - 4d = 0d)- For e: (3e + e - 4e = 0e)- Constants: (1 + 13 + 103 = 117)Wait, that's interesting. The d and e terms cancel out, leaving just 117. So the sum of the right-hand sides is 117.Now, let's compute the left-hand side:((a + 1)(3bc + 1) + (b + 1)(3ca + 1) + (c + 1)(3ab + 1))Let me expand each term:1. ((a + 1)(3bc + 1) = 3bc(a + 1) + (a + 1) = 3abc + 3bc + a + 1)2. Similarly, ((b + 1)(3ca + 1) = 3abc + 3ca + b + 1)3. And ((c + 1)(3ab + 1) = 3abc + 3ab + c + 1)Now, adding all these together:(3abc + 3bc + a + 1 + 3abc + 3ca + b + 1 + 3abc + 3ab + c + 1)Combine like terms:- For (abc): (3abc + 3abc + 3abc = 9abc)- For (ab), (bc), (ca): (3ab + 3bc + 3ca)- For a, b, c: (a + b + c)- Constants: (1 + 1 + 1 = 3)So the left-hand side becomes:(9abc + 3(ab + bc + ca) + (a + b + c) + 3)Now, we have:(9abc + 3(ab + bc + ca) + (a + b + c) + 3 = 117)Hmm, that's a bit complicated. Maybe I can factor this somehow or find a way to express it in terms of a, b, c.Wait, let me think. Maybe I can factor out 3 from some terms:(9abc + 3(ab + bc + ca) + (a + b + c) + 3 = 117)Let me write this as:(3(3abc + ab + bc + ca) + (a + b + c) + 3 = 117)Hmm, not sure if that helps. Maybe another approach.Alternatively, perhaps I can assume that a, b, c are small integers since the right-hand side is 117, which isn't too large. Let me try small values for a, b, c.Let's start by assuming a = b = c = 1.Plugging into the left-hand side:(9*1*1*1 + 3(1*1 + 1*1 + 1*1) + (1 + 1 + 1) + 3 = 9 + 9 + 3 + 3 = 24), which is much less than 117.Okay, too small. Let's try a = b = c = 2.Left-hand side:(9*2*2*2 + 3(2*2 + 2*2 + 2*2) + (2 + 2 + 2) + 3 = 72 + 3*12 + 6 + 3 = 72 + 36 + 6 + 3 = 117)Oh! That works. So a = b = c = 2.Great, so now we know a = b = c = 2.Now, let's plug these back into the original equations to find d and e.First equation:((a + 1)(3bc + 1) = d + 3e + 1)Plugging a = 2, b = 2, c = 2:((2 + 1)(3*2*2 + 1) = d + 3e + 1)Calculate:(3*(12 + 1) = d + 3e + 1)(3*13 = d + 3e + 1)(39 = d + 3e + 1)So, (d + 3e = 38) ... (1)Second equation:((b + 1)(3ca + 1) = 3d + e + 13)Again, b = 2, c = 2, a = 2:((2 + 1)(3*2*2 + 1) = 3d + e + 13)Calculate:(3*(12 + 1) = 3d + e + 13)(3*13 = 3d + e + 13)(39 = 3d + e + 13)So, (3d + e = 26) ... (2)Now, we have two equations:1. (d + 3e = 38)2. (3d + e = 26)Let me solve this system. Let's use substitution or elimination. Maybe elimination.Multiply equation (2) by 3:(9d + 3e = 78) ... (2a)Now, subtract equation (1) from (2a):(9d + 3e - (d + 3e) = 78 - 38)Simplify:(8d = 40)So, (d = 5)Now, plug d = 5 into equation (2):(3*5 + e = 26)(15 + e = 26)(e = 11)So, d = 5 and e = 11.Let me check these values in the third equation to make sure.Third equation:((c + 1)(3ab + 1) = 4(26 - d - e) - 1)Plugging c = 2, a = 2, b = 2, d = 5, e = 11:Left-hand side:((2 + 1)(3*2*2 + 1) = 3*(12 + 1) = 3*13 = 39)Right-hand side:(4*(26 - 5 - 11) - 1 = 4*(10) - 1 = 40 - 1 = 39)Perfect, both sides equal 39. So the values are consistent.Therefore, d = 5 and e = 11.Now, the question asks for (d^2 + e^2).Calculating:(5^2 + 11^2 = 25 + 121 = 146)So, the answer is 146.**Final Answer**boxed{146}